Great Western Coffee Shop

With over 2537 (or so) stations in the network, there are 6,433,832 different journey opportunities between stations. A very few of them aren't provided for (Catford to Catford Bridge and St Budeaux Ferry Road to St Budeaux Victoria Road come to mind) but there are still a massive number of journeys that must be rarely made .... Roman Bridge to Sandplace, anyone?

What does the rail industry consider special if you are travelling ...
1. ... from Invergordon to Long Buckby?
2. ... from Bognor Regis to Amersham?
3. ... from Llantwit Major to Cardiff Central?
4. ... from Lydney to Bristol Temple Meads?
5. ... from Goring on Sea to Maidstone East
6. ... from Rainham to Pitlochry
7. ... from Corkickle to Douglas
8. ... from St Andrews Road to Pilning on a Saturday
9. ... from Castleton to Cambridge
10. ... from Smitham to Redhill
11. ... from Melksham to Avoncliff
12. ... from Bristol Temple Meads to Stonehouse

A guess. Do they all involve either a walk between stations or the need to travle over part of a route twice?

3. Llantwit Major to Cardiff Central. You can go via Barry or Bridgend.

From St Andrews Road to Pilning on a Saturday. Thumb a lift ;D

I can't fathom what the selection for this quiz is based on, so this is an exploration. Many of them look like routeing test cases for journey planners, which is a staring point.

Picking Castleton (Manchester)-Cambridge to look at, I did wonder if it had the most routeings (29 sets of maps). But Cambridge-Glasgow group has 40 on its own, so for stations near Glasgow and Cambridge that are also in other routeing groups you could not doubt get that much higher.

But CAS-CBG does have some extreme fare pricing issues. As I hope you can see from the BRFares output:

Route     00312  ✠VIA MANCHESTER
Cross-London     YES
Fare Setter     IWC  AVANTI WEST COAST
FIRST CLASS
FOR  ANYTIME 1R            ◊     1ST  RETURN    ADULT: £610.40    CHILD: £305.20
FSR  OFF-PEAK 1R    2C          1ST  RETURN    ADULT: £383.80   CHILD: £191.90
FOS  ANYTIME 1S            ◊     1ST  SINGLE    ADULT: £305.20    CHILD: £152.60
FSS  OFF-PEAK 1S    2C          1ST  SINGLE    ADULT: £191.90  CHILD: £95.95
STANDARD CLASS
SOR  ANYTIME R            ◊     STD  RETURN    ADULT: £458.60    CHILD: £229.30
SVR  OFF-PEAK R    2C          STD  RETURN    ADULT: £119.80    CHILD: £59.90
SOS  ANYTIME S            ◊     STD  SINGLE    ADULT: £229.30    CHILD: £114.65
SVS  OFF-PEAK S    2C          STD  SINGLE    ADULT: £83.80    CHILD: £41.90
Route     00700  NOT VIA LONDON
Cross-London     NO
Fare Setter     IXC  CROSSCOUNTRY
FIRST CLASS
FOR  ANYTIME 1R            ◊     1ST  RETURN    ADULT: £461.30    CHILD: £230.65
FSR  OFF-PEAK 1R    2V     ◊     1ST  RETURN    ADULT: £309.60    CHILD: £154.80
FOS  ANYTIME 1S            ◊     1ST  SINGLE    ADULT: £245.80    CHILD: £122.90
STANDARD CLASS
SOR  ANYTIME R            ◊     STD  RETURN    ADULT: £184.50    CHILD: £92.25
SVR  OFF-PEAK R    2V     ◊     STD  RETURN    ADULT: £135.20    CHILD: £67.60
SOS  ANYTIME S            ◊     STD  SINGLE    ADULT: £105.90    CHILD: £52.95

For via Manchester, the ratio of anytime to off-peak return is very high: 3.83. That's not so for first class, as the first:standard ratio is also ridiculously high (3.20). The "not via London" fares also have very high First fares.

Not that the high anytime price matters, as the main restrictions on off-peak apply for travel via London, which is slower (at least for most of the day - OJP finds none). The only exception is "Not valid on trains timed to depart  Stevenage after 02:59 and before 09:19 or after 15:06 and before 19:01 (18:35 on Fridays)" - I guess that could apply.

But the "not via London" anytime return priced by Cross Country is only £184.50, so who cares?

Rational?

You are headed in the right direction, though onto a tangent which is a very interesting one.   Some of the other information / data shared in other answers also give a drift towards the commonality.

These look like routing guide easements to me.

So I'll go for 1:

"700778: Customers travelling via Dingwall to Long Buckby may travel via Rugby avoiding Northampton. This routeing point eassement [sic] overrides fares checking and operates in both directions."

You have indeed got the common connection - these are special journeys because something about each journey has something about it that needs some sort of exceptional treatment in the fare and routing system.

No doubt from Castleton-Cambrdige the uniquity is easement 700660:

While the whole concept of the NFM64 fares check is weird, in this case most non-London routes go that way so might need the easement. But I wonder if it needs another one too - for the fastest trip offered now (for Monday 20th) by the OJP, changing at Rochdale, Leeds, and Peterborough, it says "no fares available". Strange, given that at other times the same routeing does have a fare.

For 12 (from Bristol Temple Meads to Stonehouse) I don't have the answer, but I do know there are two plausible routes each round 'two sides of a square': via Cheltenham Spa and via Swindon.  So I guess this is a component of the answer.  And 4 (from Lydney to Bristol Temple Meads) looks similar: routing via the Severn tunnel or via Gloucester, and in the latter case potentially changing at Cheltenham Spa.

A bit of nostalgia.  Some 30 years ago I had a friend whose daily commute was Cheltenham Spa to Bath.  For each journey she had to decide: today shall I go via Bristol Temple Meads or Swindon?  If trains were running on time both routes were pretty much equal.  But if trains were not on time then get it wrong and there could be a long wait en route.  The right answer would depend partly on how the first-leg trains were running and even more so on how well the second-leg trains on each route were running, because this affected making or missing the connections.  But before the days of RTT one had to depend on fragmentary information from station staff about how well the distant services were running.  She suffered quite a number of missed connections when the alternative route would have provided a much faster journey that day.  Consequently she had quite frequent contact with the BR complaints department.  The point of this anecdote is that she was such a frequent customer of the department that one year they sent her a Christmas card!

10. ... from Smitham to Redhill

This one is odd, as Smitham station was renamed Coulsdon Town ten years ago. So the journey would now involve routeing via a time warp.

But there is an anomaly (you could call it a typo) in doubleback easement 700321:

This applies to all the stations on the Tattenham Corner branch, allowing a change at East Croydon rather than Purley. Except it misses out Coulsdon Town (the new Smitham) and substitutes Coulsdon South. Whether and when that longer double back would be really needed I'm not sure, but today with so many cancellations there are some big gaps without it.

What have the journey planners been fed with? Well, the OJP won't even sell a single ticket (SDS, any permitted route) from Reedham to Redhill, which is listed in the easement, for an itinerary via East Croydon. It insists you need two tickets at £4.60 each rather than one. So what it says about CDN-RDH isn't going to tell us anything.